Abstract:
For the generating function of static correlators of the third components of spins in the XX Heisenberg model, we derive a new representation given by a combination of Gaussian functional integrals over anticommuting variables. A peculiarity of the resulting functional integral is that a part of the integration variables depend on the imaginary time automorphically: these variables are multiplied by a certain complex number under the shift of the imaginary time by the period. The other variables satisfy the standard boundary conditions of the fermionic/bosonic type. Functional integration results are represented as determinants of matrix operators. We finally evaluate the generating function of correlators and the partition function of the model in the zeta-function regularization. The consistency of the suggested functional definition is confirmed by calculating several correlation functions of the third components of spins at a nonzero temperature.
Keywords:
functional integration, XX Heisenberg model, correlators, generalized zeta function.
Citation:
K. L. Malyshev, “Functional Integration with an “Automorphic” Boundary Condition and Correlators of Third Components of Spins in the XX Heisenberg Model”, TMF, 136:2 (2003), 285–298; Theoret. and Math. Phys., 136:2 (2003), 1143–1154
\Bibitem{Mal03}
\by K.~L.~Malyshev
\paper Functional Integration with an ``Automorphic'' Boundary Condition and Correlators of Third Components of Spins in the $XX$ Heisenberg Model
\jour TMF
\yr 2003
\vol 136
\issue 2
\pages 285--298
\mathnet{http://mi.mathnet.ru/tmf221}
\crossref{https://doi.org/10.4213/tmf221}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2025377}
\zmath{https://zbmath.org/?q=an:1178.82017}
\transl
\jour Theoret. and Math. Phys.
\yr 2003
\vol 136
\issue 2
\pages 1143--1154
\crossref{https://doi.org/10.1023/A:1025070022585}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000185531200008}
Linking options:
https://www.mathnet.ru/eng/tmf221
https://doi.org/10.4213/tmf221
https://www.mathnet.ru/eng/tmf/v136/i2/p285
This publication is cited in the following 4 articles:
C Malyshev, N M Bogoliubov, “Spin correlation functions, Ramus-like identities, and enumeration of constrained lattice walks and plane partitions”, J. Phys. A: Math. Theor., 55:22 (2022), 225002
N. M. Bogolyubov, K. L. Malyshev, “Ising limit of a Heisenberg XXZ magnet and some temperature correlation functions”, Theoret. and Math. Phys., 169:2 (2011), 1517–1529
N. M. Bogoliubov, K. Malyshev, “The correlation functions of the XXZ Heisenberg chain in the case of zero or infinite anisotropy, and random walks of vicious walkers”, St. Petersburg Math. J., 22:3 (2011), 359–377
K. L. Malyshev, “The condition of quasi-periodicity in imaginary time as a constraint at the functional integration and the time-dependent ZZ-correlator of the XX Heisenberg magnet”, J. Math. Sci. (N. Y.), 136:1 (2006), 3607–3624
Citation in format AMSBIB
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